All Problems Constant Magnetic Field, Magnetics
Problem 3.237
A very long straight solenoid has a cross-section radius \(R\) and \(n\) turns per unit length. A direct current \(I\) flows through the solenoid. Suppose that \(x\) is the distance from the end of the solenoid, measured along its axis. Find: (a) the magnetic induction \(B\) on the axis as a function of \(x\); draw an approximate plot of \(B\) vs the ratio \(x / R\); (b) the distance \(x_{0}\) to the point on the axis at which the value of \(B\) differs by \(\eta=1 \%\) from that in the middle section of the solenoid
3.237. (a) \(B=1 / 2 \mu_{0} n I\left(1-x / \sqrt{x^{2}+R^{2}}\right),\) where \(x>0\) outside the solenoid and \(x<0\) inside the solenoid; see Fig. \(23 ;\) (b) \(x_{0}=\) \(=R(1-2 \eta) / 2 \sqrt{\eta(1-\eta)} \approx 5 R\)